Explicit construction of Hadamard states for Quantum Field Theory in curved spacetimes

TL;DR

This thesis explicitly constructs Hadamard states for quantum fields in various curved spacetimes, including static, cosmological, and black hole backgrounds, demonstrating their properties and existence without relying on symmetry invariance.

Contribution

It provides new explicit constructions of Hadamard states in non-symmetric and complex spacetimes, including cosmological and Schwarzschild-de Sitter cases.

Findings

Constructed Hadamard states in static and cosmological spacetimes.

Demonstrated existence of non-invariant Hadamard states depending on smooth functions.

Built a non-thermal Hadamard state in Schwarzschild-de Sitter spacetime.

Abstract

In this Thesis we will explicitly construct Hadamard states with different features. In the two first cases, the states will be constructed in static and in cosmological spacetimes. In one of the cases, the states will be constructed in such a way as to minimize the expectation value of the energy-momentum tensor of a free massive scalar field, multiplied by a suitable smooth function of compact support, in a cosmological spacetime. The final form of the state clearly reduces to the vacuum in a static spacetime. Besides, we will show that these are Hadamard states. They do not contradict the above mentioned nonexistence of a unique prescription because they depend on the chosen smooth function. In the other case, we will calculate the positive part of the spectral decomposition of the commutator function in a specific subset of the spacetime and will show that, in static and in cosmological spacetimes, the kernel of this function, once again multiplied by a suitable smooth function of compact support, corresponds to the kernel of Hadamard states. In both cases, the construction of the states does not rely on invariance under the action of a group of symmetries. Such a group may not even exist, in these cases. In the last case we will construct a state in the Schwarzschild-de Sitter spacetime, which describes a universe with a black hole and a cosmological constant. This state is invariant under the action of the group of symmetries of this spacetime, although it is not defined in its whole Kruskal extension. In this way, we avoid the situation in which the authors of (KayWald91) showed that a Hadamard state cannot exist. We will show that the state we construct is Hadamard and that it is not thermal, differently from the case of states constructed in spacetimes containing only one event horizon (DappiaggiMorettiPinamonti09,GibbonsHawking77).